---- - Let $E \xrightarrow{\pi}B$ be a [[vector bundle|smooth vector bundle]] with typical fiber $\mathbb{R}^{m}$ over a [[smooth manifold]] $B$ of dimension $n$. - Let $U \subset B$ simultaneously a denote a [[coordinate chart|coordinate neighborhood]] on $B$ and a [[vector bundle|trivializing neighborhood]] for $E$. - Fix coordinates $(\boldsymbol x^{k})=(x^{1},\dots,x^{n})$ on $U$, and $(\boldsymbol a^{j})=(a^{1},\dots,a^{m})$ on the typical fiber $\mathbb{R}^{m}$. With this notation, we have further - A [[vector bundle|local trivialization]] $\Phi_{U}:\pi ^{-1}(U) \xrightarrow{\sim} U \times \mathbb{R}^{m}$. - For any $p \in \pi ^{-1}(U) \subset E$, the [[tangent space at a point of a smooth manifold|tangent space]] at $p$ looks like $T_{p}E=\text{span}\left( \frac{ \partial }{ \partial x^{k} } |_{p} , \frac{ \partial }{ \partial a^{j} } |_{p} \right)_{j \in [m], k \in [n]}$ in the local trivialization. The '$|_{p} will often be omitted. - Let $b:=\pi(p)$. Then $E_{b} \subset E$ is an [[embedded submanifold]] ([[regular value theorem|RVT]]) with $\begin{align} T_{p}E_{b}&= \text{ker}(d \pi |_{p}: \text{span}\left( \frac{ \partial }{ \partial x^{k} } |_{p}, \frac{ \partial }{ \partial a^{j} } |_{p} \right) \to \text{span}\left( \frac{\partial{}}{ \partial x^{k} } |_{p} \right)) \\ &=\text{span}\left( \frac{ \partial }{ \partial a^{j} } |_{p} \right)_{j \in [m]}. \end{align}$ Note that $\dim T_{p} E_{b}=m$. > [!definition] Definition. ([[field of horizontal subspaces]]) > [[linear subspaces as intersections of kernels of 1-forms|Recall]] that any $n$-dimensional subspace of $T_{p}E \cong \mathbb{R}^{m+n}$ is $\bigcap_{i=1}^{m} \text{ker } \theta^{i}$ for some [[linearly independent]] elements $\theta^{i} \in (\mathbb{R}^{n+m})^{*}$ — in other words, is the solution set of a system of $m$ independent linear relations.[^7] With this, we can represent any [[horizontal subspace]] at $p$ as the simultaneous kernel $S_{p}=\{ v \in T_{p}E : \theta^{i}_{p} (v) =0 \text{ for all } i=1,\dots,m \}$ where, letting $T_{p}^{*}E$ denote the [[cotangent space]] of $E$ at $p$ and employing summation convention, $\theta^{i}_{p}$ is a [[dual vector space|linear functional]] > $\theta^{i}_{p}=f^{i}_{k} \ dx^{k} + g_{j}^{i} \ da^{j} \in T_{p}^{*}E$ for some scalars $f^{i}_{k=1,\dots,n}, g^{i}_{j=1,\dots,m} \in \mathbb{R}$. > > We call a map $p \mapsto S_{p}$ a **smooth field of horizontal spaces** if the functions $f_{k}^{i}(p)$ and $g_{j}^{i}(p)$ are smooth. ![[Pasted image 20250402205539.png]] ^definition [^7]: In coordinate matrix terms: $S_{p}=\operatorname{ker } \boldsymbol \Theta$ for some $\boldsymbol \Theta \in \mathbb{R}^{m \times(n+m)}$ having linearly independent rows. This perspective is expanded upon below. > [!proposition] Preferred Coordinates. > We can replace $\theta$ as written above with something better. Every smooth field of horizontal subspaces $S=\{ S_{p}: p \in E \}$ locally (i.e., in a local trivialization; identify $p=(x,a) \in U \times \mathbb{R}^{m}$) may be expressed as a simultaneous kernel[^1] $S_{p}=\bigcap_{i=1}^{m} \text{ker }\theta^{i}_{p} \text{ with } \theta^{i}_{p}=da^{i}+e_{k}^{i}(x,a) \ dx^{k}$ > for some $e_{k}^{i} \in C^{\infty}(U \times \mathbb{R}^{m})$ uniquely determined by the local trivialization. > > ^proposition > [!proof] Proof of Existence of Preferred Coordinates. > Fix $p \in E$, identified with $(x,a) \in U \times \mathbb{R}^{m}$ under the local trivialization $\Phi_{U}$. We know that $c \in T_{p}E$, $c \neq 0$, is [[vertical subspace|vertical]] iff $c=c^{j} \frac{ \partial }{ \partial a^{j} }$ for some $c^{j=1,\dots,m} \in \mathbb{R}$ not all zero; thus there exists $i \in [m]$ for which $\theta^{i}\left( c^{j} \frac{ \partial }{ \partial a^{j} } \right)= f_{k}^{i} c^{j} \ \cancel{dx^{k} \left( \frac{ \partial }{ \partial a^{j} } \ \right)}^{=0} + g_{j}^{i} c^{j} \ da^{j} \left( \frac{ \partial }{ \partial a^{j} } \right) \neq 0$ for all nonzero $c \in T_{v_{p}}E$. It follows that $g_{j}^{i}c^{j}=0$ iff $c=0$, that is, the vectors $g^{}_{j}$ are [[linearly independent]]. Concatenate them into an invertible $m \times m$ [[matrix]] $(g_{j}^{i})_{i,j=1,\dots,m}$ with [[inverse matrix|inverse]] $(h_{j}^{i})_{i,j=1,\dots,m}$ from which the "improved $\theta$s" are obtained as $\tilde{\theta}_{p}^{i}:= h_{j}^{i} \theta_{p}^{j}=da^{i}+e_{k}^{i} \ dx^{k},$ where $e_{k}^{i}=h_{j}^{i}f_{k}^{j}$. I find that the matrix version below of this proof tends to balance nicer in my head. > ^proof > [!proof] Matrix Version: Proof of Existence of Preferred Coordinates. > Here is the above discussion from a matrix perspective: in [[matrix|matrix form]], $\theta^{i}=f_{k}^{i} \, dx^{k}+ g^{i}_{j}\, da^{j}$ is $\boldsymbol \theta^{i}=\begin{bmatrix} f^{i}_{1} & \cdots f^{i}_{n} & g^{i}_{1} & \cdots & g^{i}_{m} \end{bmatrix} \in \mathbb{R}^{1 \times (n+m)}.$We stack into a [[matrix]] $\boldsymbol \Theta \in \mathbb{R}^{m \times (n+m)}$, $\boldsymbol \Theta= \begin{bmatrix} > \textemdash & \boldsymbol \theta^{1} & \textemdash \\ > & \vdots & \\ > \textemdash & \boldsymbol \theta^{m} & \textemdash > \end{bmatrix}= \begin{bmatrix} > f^{1}_{1} & \cdots & f^{1}_{n} & g^{1}_{1} & \cdots& g_{m}^{1} \\ > \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ > f_{1}^{m} & \cdots & f_{n}^{m} & g_{1}^{m} & \cdots & g_{m}^{m} > \end{bmatrix}=: \begin{bmatrix} > \boldsymbol F_{m \times n} & \boldsymbol G_{m \times m}. > \end{bmatrix}$ > By construction, the horizontal subspace $S$ equals $\operatorname{ker }\boldsymbol \Theta$. The [[vertical subspace]] $T_{v_{p}}E$ equals the collection of coordinate vectors of the form $\begin{bmatrix}\boldsymbol 0 \\ \boldsymbol c\end{bmatrix}$. Since $\underbrace{ S }_{ \operatorname{ker } \boldsymbol \Theta } \cap T_{v_{p}}E=\{ 0 \}$, $\boldsymbol \Theta \begin{bmatrix}\boldsymbol 0 \\ \boldsymbol c\end{bmatrix}=\boldsymbol 0$ implies $\boldsymbol c=\boldsymbol 0$. Thus the columns of $\boldsymbol G_{m \times m}$ are [[linearly independent]]; write $\boldsymbol H_{m \times m}:= \boldsymbol G_{m \times m}^{-1}$. Define $\begin{align} > \widetilde{\boldsymbol \Theta} :=\boldsymbol H \boldsymbol \Theta > = \boldsymbol H_{m \times m} \begin{bmatrix} > \boldsymbol F_{m \times n} & \boldsymbol G_{m \times m} > \end{bmatrix} =\begin{bmatrix} > (\boldsymbol H \boldsymbol F)_{m \times n} & \boldsymbol I_{m \times m} > \end{bmatrix} > \end{align}$ > Since $\boldsymbol H$ and $\boldsymbol I$ are invertible, $\operatorname{ker }\widetilde{\boldsymbol \Theta}=\operatorname{ker } \boldsymbol \Theta$, meaning that $\widetilde{\boldsymbol \Theta}$ also defines the horizontal subspace $S$. Further writing $\widetilde{\boldsymbol \Theta}=\begin{bmatrix} > \left( { h^{i}_{j}f_{k}^{j} } \right)_{k \in[n]}^{i \in [m]} & \boldsymbol I_{m \times m} > \end{bmatrix} > = \begin{bmatrix} > (e^{i}_{k}) _{k \in [n]}^{i \in [m]} & \boldsymbol I_{m \times m} > \end{bmatrix},$ > we have that $\widetilde{\boldsymbol \Theta}$ is the coordinate representation of the forms $\widetilde{\theta}^{i}= e^{i}_{k} \, dx^{k} + \, da^{i}.$ > ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```